The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

learn more… | top users | synonyms (1)

4
votes
1answer
54 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
1
vote
0answers
21 views

Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement: Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and ...
1
vote
0answers
49 views

Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
3
votes
1answer
35 views

Order zero maps in matrix algebra

Let $a$ and $b$ are two elements in a $C^*$algebra $A$. We say $a\perp b$ if $ab=ba=a^*b=ab^*=0$. We say a completely positive map $\phi: A \rightarrow B$ is of order zero if for any positive elements ...
4
votes
1answer
123 views

Where does the double commutant theorem fails for $AW^*$-algebras?

Commutative $AW^*$-algebra are characterized as those $C^*$-algebras such that their space of projections is a complete boolean algebra (see http://en.wikipedia.org/wiki/AW*-algebra). Von Neumann ...
4
votes
1answer
70 views

Existence of idempotents versus existence of projections in a C*-algebra

Let $\mathcal{A}$ be any C*-algebra. Suppose $x\in\mathcal{A}$ is idempotent, with $x\neq 0$ and $x\neq 1$. Does it follow that $\mathcal{A}$ admits nontrivial projections? Clearly, when $x$ is ...
1
vote
3answers
49 views

Non-closed ideals in $C^*$-algebras

What is an example of an ideal in a commutative $C^*$-algebra that is not closed? If by chance every ideal in a commutative $C^*$-algebra is closed, how about in non-commutative $C^*$-algebras? ...
2
votes
1answer
50 views

Norms on unitization of nonunital Banach algebra

Let $A$ be a nonunital Banach algebra and denote by $A^+$ the unitization of $A$. One commonly used Banach algebra norm on $A^+$ is given by $||(a,\lambda)||=||a||+|\lambda|$ (where $a\in ...
1
vote
0answers
19 views

C^* algebra generated by a C^* algebra and a group

In this article: http://ac.els-cdn.com/S0022123606001856/1-s2.0-S0022123606001856-main.pdf?_tid=ddbd529e-b857-11e4-9f30-00000aab0f27&acdnat=1424365005_7c9a5ac14284f6258613e8a6cb8a9482 "Spectral ...
2
votes
1answer
51 views

when a crossed product group is inner amenable

Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not. For example, when $\alpha$ is trivial, ...
0
votes
0answers
90 views

Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
2
votes
1answer
26 views

Are ideals generated by separable subspaces separable?

Suppose that $X$ is a compact Hausdorff space and take a sequence $(f_n)$ in $C(X)$ such that the ideal generated by $(f_n)$ is proper. Must this ideal be separable as a Banach space? It looks to me ...
1
vote
1answer
37 views

C* Algebra Positivity

STATEMENT: This is a proof from one of Qiaochu's notes on $C^*$ algebras. Proof: Let A be a $C^*$ algebra.We now want to show that for any $c\in A$ we have $c^*c\geq 0$. Suppose otherwise.We know ...
4
votes
0answers
35 views

Infinite dimensional C*-algebra contains infinite dimensional commutitive subalgebra

I was reading a paper which mentioned without proof that every infinite-dimensional $C$* algebra has an infinite-dimensional commutative $C$* subalgebra. Thinking about it for 10 minutes, I didn't ...
1
vote
0answers
27 views

Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into ...
0
votes
1answer
48 views

A question about induced $C^\ast$-algebra

Recently, I read the book Crossed Products of C*-algebras, and meat a question. The question is how to prove $\mathrm{Ind}_c(A,\alpha)$ is dence in $\mathrm{Ind}(A,\alpha)$. On the page 102, the ...
1
vote
0answers
32 views

Group $C^*$-algebra elements as limit of self-adjoint integrable functions

Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra. I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions ...
0
votes
1answer
58 views

Partial Isometries: Characterization

Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: $$W^*W,WW^*$$ One derives a partial isometry: ...
2
votes
1answer
34 views

Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
1
vote
1answer
20 views

A symmetric algebra that is not a C* algebra

Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...
-1
votes
1answer
42 views

Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
1
vote
1answer
27 views

States: KMS-Condition

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state $\omega$. Does it suffice to have on a dense set the KMS-condition: ...
1
vote
0answers
31 views

Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
0
votes
0answers
233 views

Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
-1
votes
1answer
48 views

Tensor Products of C*-Algebras

If A, B are C*-algebras, show that there exists a unique $*-isomorphism $ $ ‎\theta‎‎: A ‎\otimes‎_{*}‎‎ B ‎\longrightarrow‎ B ‎\otimes‎_{*}‎‎ A $ such that $\theta( a \otimes‎_{*} b) = b ...
3
votes
1answer
59 views

Question about $C_0(X)$-algebras and $C_b(X)$.

Let $X$ be a locally compact Hausdorff space. Denote by $C_0(X)$ its C*-algebra of continuous functions that vanish on infinity and by $C_b(X)$ its C*-algebra of bounded functions. Now, let $A$ be a ...
0
votes
0answers
34 views

Spectral decomposition of a an Operator on $\mathcal{L}(l^2(\mathbb{N})$

I have an exercise, where I do not really know how to solve it. Let $\{\lambda_n\}_n$ be a counting of the rational numbers in $[0,1]$ and define a bounded self-adjoint operator $T$ on ...
2
votes
1answer
63 views

Self adjoint operator has spectrum around 0 and 1

I need to prove the following statement: Let $\mathcal{A}$ be a unital $C^*-$Algebra, $A$ a self-adjoint element and $P$ a projection, so $P^2=P=P^*$. Let $\delta :=\|P-A\|$. I want to prove that ...
0
votes
0answers
50 views

Power of positive operator

Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in ...
1
vote
0answers
26 views

States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
2
votes
0answers
28 views

Proving the unitary relation of ensemble decompositions

I apologize if this is better suited for physics.stackexchange; looking at previously asked questions it seems as if this would be a good fit here, as the arguments are mostly mathematical. In my ...
1
vote
0answers
40 views

States: Approximate Unit

Given a C*-algebra $\mathcal{A}$. Consider a state: $\omega\geq0$ Especially one has: $\sup\omega(E^2)=\|\omega\|$ Can it actually fail to be a proper limit? The problem is that the square is not ...
-1
votes
1answer
80 views

Positive Elements: Norm (Decomposition)

Given a C*-algebra $\mathcal{A}$. Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$ Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$ ...
1
vote
1answer
54 views

States: Positivity vs. Continuity

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$. Consider a ...
1
vote
1answer
39 views

States: Extension

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit ...
2
votes
1answer
57 views

An elementary perturbation result in C*-algebra

The following question was raised when I read a papaer "MF actions and K-theoretic dynamics". In one of the proofs in that paper, the author utilized a so called "elementary perturbation result in ...
4
votes
0answers
99 views

On the exponential form of a unitary matrix

A unitary matrix $U \in \large C_{n,n}$ can always be written in the exponential form: $U = e ^{iA}$ (1) where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary ...
-1
votes
1answer
37 views

Approximate Identity: Projections

Problem Given a C*-algebra $\mathcal{A}$. Denote the positive open unit ball by: $\mathcal{B}^+$ Then it has an approximate identity: ...
2
votes
1answer
32 views

Positive Elements: Decomposition?

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Then every selfadjoint element decomposes into positives ones: $$A=A^*:\quad A=\frac12(|A|-A)-\frac12(|A|-A)\quad\left(\frac12(|A|\pm ...
0
votes
1answer
13 views

Dynamics: EQ-States vs. NESS-States

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state that relaxes towards equilibrium: $$\omega_T(A):=\omega\circ\tau^T(A)\stackrel{T\to\infty}{\to}\omega_\infty(A)$$ Then it ...
3
votes
1answer
45 views

Set-theoretic questions about the definitions of crossed-product $ C^{*} $-algebras and group $ C^{*} $-algebras.

In his book Crossed Products of $ C^{*} $-Algebras, Dana P. Williams defines the crossed product of a $ C^{*} $-algebra $ A $ by a locally compact group $ G $ as the completion of $ {C_{c}}(G,A) $ ...
1
vote
1answer
47 views

A sufficient condition for a strictly positive linear map

Let $\mathcal{A}$ and $\mathcal{B}$ be ${\rm C}^*$-algebras, $\psi: \mathcal{A} \to \mathcal{B}$ a linear positive map (i.e. $a \ge 0 \Rightarrow \psi(a) \ge 0$) and $p \in A$ projection such that ...
0
votes
1answer
41 views

A question about the extension of homomorphism

Here is a quotation of a book "C*-algebras and Finite-Dimensional Approximations" P275 Let $A$ be a C*-algebra. Suppose $I\triangleleft A$ be a closed ideal. If the representation ...
0
votes
1answer
45 views

trace identities for the functional calculus

I'm sorry if this is a trivial question, but I cannot convince myself of why $\text{Tr}\,f(EFE)=\text{Tr}\,f(FEF)$ for projections $E$ and $F$ on a Hilbert space and a (say, continuous or Borel) ...
2
votes
0answers
40 views

about ultrapowers

Let $\mathcal{R}$ denote the hyperfinite type $II_{1}$ factor, with $\mathcal{R}^{\omega}$ the ultrapower of $\mathcal{R}$, in respect to some ultrafilter $\omega$ on $\mathbb{N}$. I'm reading a paper ...
1
vote
1answer
36 views

A question about the definition of group $C^\ast$-algebra

Let $G$ be a local compact group, then group $C^\ast$-algebra of $G$ is defined as the completion of $C_c(G)$ with respect to some norm. By now, I have seen three norms. $\|f\|=\sup\|\pi(f)\|$, ...
1
vote
0answers
20 views

Prove a factor is III$_{\lambda}$ type

This question is from the Sunder's book An Invitation to von Neumann Algebras Ex 4.2.13 Let M be a semifinite factor with fns trace ${\tau}$. Let ${\theta}$ be an automorphism of M such that ...
3
votes
1answer
45 views

A question about essential ideal

Let $I$ be a nonunital C*-algebra and $I\subset B(H)$ be any nondegenerate representation and define $$M(I)=\{T\in B(H): Tx\in I~and ~xT\in I, ~for ~all~ x\in I\}.$$ Then, how to prove $I$ ...
5
votes
1answer
54 views

The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a ...
2
votes
1answer
80 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...